Graphs in Practical Situations | IGCSE Maths Revision

Syllabus coverage

Use and interpret graphs in practical situations, including travel graphs and conversion graphs
Interpret the gradient of a straight-line graph as a rate of change
Draw graphs from given data, for example a distance-time graph to represent a journey
Extended only: estimate and interpret the gradient of a tangent at a point
Extended only: apply the idea of rate of change to simple kinematics involving distance-time and speed-time graphs, acceleration and deceleration
Extended only: calculate distance travelled as the area under a speed-time graph, where the areas involve linear sections of the graph only

A straight-line graph with its gradient and intercept marked. In a practical situation the gradient is the rate of change and the intercept is the value of the vertical quantity when the horizontal one is zero.

What the axes and the shape are telling you

Every practical graph starts with the same three questions: what is on each axis, what units are they in, and what does one point mean? On a conversion graph the two axes are two different units for the same quantity, so the point $(5, 8)$ reads '5 miles is the same as 8 kilometres'. On a travel graph one axis is time and the other is distance or speed, so each point is a moment in a journey.

The shape carries the meaning. A straight line means the rate is constant, so the same amount is being added for every unit of time or distance. A steeper line means a bigger rate. A line that flattens out means the rate has dropped to nothing.

Read a value off a graph by going up from the horizontal axis to the line and then across to the vertical axis, and read it to the accuracy the grid allows. If the grid is marked every 10 minutes, an answer of 35 minutes is honest and 34.7 minutes is not: the graph cannot tell you more than it shows.

A straight line: the rate of change is constant
A steeper line: a larger rate of change
A horizontal line: nothing is changing
A line sloping back down towards the time axis: the quantity is decreasing

Write the units next to the gradient every time. They tell you whether you have found a speed, a cost, a conversion factor or a rate of filling, and they are usually worth a mark on their own.

Gradient is a rate of change

The gradient of a straight line measures how much the vertical quantity changes for every unit of the horizontal one. Find it from two points on the line: the change in $y$ divided by the change in $x$ . A line through $(0, 2)$ and $(5, 12)$ has a gradient of $\frac{12 - 2}{5 - 0} = 2$ , which means the vertical quantity grows by 2 for every 1 across.

In a practical situation that number is a rate, and the units come from the axes. A distance-time graph with kilometres and hours has a gradient in kilometres per hour. A cost-distance graph has a gradient in dollars per kilometre. A depth-time graph for a filling tank has a gradient in centimetres per minute. Naming those units is part of the answer.

The intercept is the value of the vertical quantity when the horizontal one is zero, and in context it almost always has a meaning: the fixed charge on a bill, the distance already travelled when the timing started, the depth of water in a tank before the tap was turned on.

Gradient = the change in $y$ divided by the change in $x$
On a distance-time graph, the gradient is the speed
On a speed-time graph, the gradient is the acceleration
On a conversion graph, the gradient is the conversion factor

The gradient is how much y changes for each 1 across; the intercept is where the line meets the y-axis. On the line y = 2x - 3 the gradient is 2 and the intercept is -3.

Distance-time graphs

On a distance-time graph the gradient is the speed: distance divided by time. A car that covers 60 km in 45 minutes has travelled at $60 \div 0.75 = 80$ km/h, because 45 minutes is 0.75 hours. Either work in hours from the start, or work in kilometres per minute and multiply by 60 at the end. Mixing the two is what produces the wrong answer.

The shape of the line is the story of the journey. A steeper section is a faster part of the journey. A horizontal section means the distance is not changing, so the object is stationary. A section sloping back down towards the time axis means the object is returning towards its starting point.

The average speed for a whole journey is the total distance divided by the total time, which is the gradient of the straight line joining the first and last points of the graph. It is not the average of the separate speeds, because the object spends different amounts of time at each speed.

Section of a distance-time graph	What it means	Gradient
Straight line sloping up	Moving away at a steady speed	the speed
Horizontal line	Stationary: the distance is not changing	zero
Straight line sloping down	Returning towards the start	the speed, written as a negative number
Curve	The speed is changing	the speed at that moment, from the tangent

A distance-time graph in a 0580 question measures distance from the start, so coming back is a line heading down towards zero rather than a line below the axis.

Speed-time graphs: acceleration and distance

On a speed-time graph the two jobs swap over. The gradient is now the acceleration, the rate at which the speed changes, and the area between the graph and the time axis is the distance travelled. Getting those two the wrong way round is the most expensive mistake in this topic, because the two methods give answers of similar size.

Acceleration is measured in metres per second squared, m/s $^{2}$ . A car that reaches 12 m/s from rest in 8 seconds has an acceleration of $12 \div 8 = 1.5$ m/s $^{2}$ . A line sloping down means the object is slowing down, which the syllabus calls deceleration, and the gradient is then negative.

To find the distance, split the area under the graph into triangles, rectangles and trapezia, work out each area and add them. Every section in a 0580 question is a straight line, so the shapes are always ones you know. A trapezium section has area $\frac{1}{2} (a + b) h$ , where $a$ and $b$ are the two speeds at the ends of the section and $h$ is the time it lasts.

A horizontal line on a speed-time graph means the speed is constant, so the acceleration is zero. It does not mean the object is stationary: it is still moving, and the area under that section is still distance.

Section of a speed-time graph	What it means	What to work out
Straight line sloping up	Speeding up steadily	gradient = acceleration, area = distance
Horizontal line	Constant speed, zero acceleration	area = distance
Straight line sloping down	Slowing down steadily	gradient = deceleration, area = distance
Area under the whole graph	The whole journey	total distance travelled, in the units of the axes

The areas are in metres only if the axes are metres per second and seconds. Write the units of the area as you work it out: speed in m/s multiplied by time in s gives metres, and a triangle's area has the same units as the rectangle it half-fills.

Conversion graphs

A conversion graph shows two units for the same quantity, one on each axis: miles against kilometres, dollars against pounds, kilograms against pounds. Because zero of one unit is zero of the other, the line passes through the origin, and the two quantities are in direct proportion.

One point on the line gives the conversion factor, which is the gradient. If the line passes through $(5, 8)$ , then 5 miles is 8 kilometres and 1 mile is $8 \div 5 = 1.6$ kilometres. Multiply by 1.6 to go from miles to kilometres, and divide by 1.6 to come back the other way.

Reading a conversion graph is often quicker than calculating: go up from the value you know to the line, then across to the other axis. Cambridge accepts a value read from the graph to the nearest division, but you must say which unit you are giving, because the answer is meaningless without it.

The line passes through $(0, 0)$ , because 0 of one unit is 0 of the other
The gradient is the number of the vertical unit in one of the horizontal unit
A graph of cost against distance is not a conversion graph, but its gradient is the cost per unit and its intercept is the fixed charge
A graph of distance against time does not pass through the origin unless the timing started when the object was at the start

Drawing a graph from given data

The syllabus asks you to draw graphs from given data, and the marks are for the axes and the shape rather than for neatness. Choose a scale that uses most of the grid, label both axes with the quantity and the unit, and mark the points with small crosses so they are still visible under the line you draw.

Join the points with straight line segments unless the relationship itself is a curve. Travel data is a sequence of steady speeds, so the graph is a series of straight pieces, and rounding the corners turns a correct graph into a wrong one. Plot a smooth curve through the points only when the relationship is continuous, as it is for a quadratic or a reciprocal.

Mark a few numbers on each axis rather than every division: 0, 20, 40 and so on. A scale that goes up in steps of 3 is legal but wastes time and invites plotting errors.

Extended candidates also estimate the gradient of a curve at a point by drawing a tangent. Draw the tangent as long as the grid allows, touching the curve at the point without cutting through it, then read two points on the tangent as far apart as possible. Two points close together magnify any drawing error.

Axes: quantity and unit on both, with a scale that fills the grid
Points: small crosses, plotted accurately
Joining: straight segments for data, a smooth curve for a function
Labels: name the part of the graph that the question asks about

A sketch of a journey graph is worth doing even when it is not asked for. Marking the stages on a rough graph turns a wordy question into four short calculations, and it stops you adding the wrong times together.

Key terms you must be able to define

Travel graph: A graph of a journey, with time on the horizontal axis and either distance or speed on the vertical axis.

Conversion graph: A straight-line graph showing two units for the same quantity, one on each axis. It passes through the origin because zero of one unit is zero of the other.

Distance-time graph: A travel graph with distance on the vertical axis. The gradient is the speed and a horizontal line means the object is stationary.

Speed-time graph: A travel graph with speed on the vertical axis. The gradient is the acceleration and the area under the graph is the distance travelled.

Gradient: How much the vertical quantity changes for each unit of the horizontal one: the change in y divided by the change in x.

Rate of change: How quickly one quantity changes as another changes. The gradient of a graph is its rate of change, and its units come from the axes.

Intercept: The value of the vertical quantity when the horizontal one is zero, which is where the line crosses the vertical axis.

Speed: The distance travelled in unit time. On a distance-time graph it is the gradient.

Average speed: The total distance of a journey divided by the total time taken, which is the gradient of the line joining the first and last points of the graph.

Acceleration: The rate at which speed changes, measured in metres per second squared. On a speed-time graph it is the gradient.

Deceleration: A decrease in speed, shown on a speed-time graph by a line sloping down towards the time axis.

Stationary: Not moving. On a distance-time graph it is a horizontal line, because the distance is not changing.

Area under a graph: The region between the line and the horizontal axis. On a speed-time graph its area is the distance travelled.

Tangent: A straight line that touches a curve at one point without crossing it. Its gradient is the rate of change at that point.

In context

Cambridge sets 0580 in real situations - a recipe, a bank statement, a map, a garden, a graph of a car's journey. These show how the mathematics appears in a question, then you write your own working for the numbers you are given.

A conversion graph

A conversion graph for miles and kilometres is a straight line through the origin and the point $(5, 8)$ . One mile is $8 \div 5 = 1.6$ kilometres, so 12 miles is $12 \times 1.6 = 19.2$ kilometres, and 20 kilometres is $20 \div 1.6 = 12.5$ miles.

Gradient as a rate

A tank is being filled. Its depth-time graph is a straight line through $(0, 2)$ and $(5, 12)$ , with the depth in centimetres and the time in minutes. The gradient is $(12 - 2) \div (5 - 0) = 2$ , so the water rises 2 cm per minute, and the intercept of 2 cm is the depth before the tap was turned on.

A distance-time journey

A cyclist rides 12 km in 30 minutes, rests for 15 minutes, then rides back to the start in 15 minutes. The graph rises from $(0, 0)$ to $(30, 12)$ , is horizontal from 30 to 45 minutes, then falls to $(60, 0)$ . The speed out is $12 \div 0.5 = 24$ km/h, the speed back is $12 \div 0.25 = 48$ km/h, and the average speed for the whole hour is $24 \div 1 = 24$ km/h.

Area under a speed-time graph

A car accelerates from rest to 12 m/s in 8 seconds, travels at 12 m/s for 20 seconds, then slows to rest in 6 seconds. The distance is the area under the graph: $\frac{1}{2} \times 8 \times 12 = 48$ m, then $12 \times 20 = 240$ m, then $\frac{1}{2} \times 6 \times 12 = 36$ m, giving $48 + 240 + 36 = 324$ m.

Interpreting an intercept

A taxi fare graph is the straight line $y = 1.8 x + 2.5$ , where $y$ is the fare in dollars and $x$ is the distance in kilometres. The gradient 1.8 is the cost per kilometre and the intercept 2.5 is the fixed charge, so a 10 km journey costs $1.8 \times 10 + 2.5 = 20.5$ dollars.

Flashcards

Click a card to flip it. Mark honestly — the tracker uses this.

Q1What does the gradient of a distance-time graph represent?

The speed: the change in distance divided by the change in time.

Q2What does the gradient of a speed-time graph represent?

The acceleration, in metres per second squared. A negative gradient is a deceleration.

Q3What does the area under a speed-time graph represent?

The distance travelled. Split it into triangles, rectangles and trapezia and add the areas.

Q4What does a horizontal line on a distance-time graph mean?

The object is stationary: the distance is not changing, so the speed is zero.

Q5What does a horizontal line on a speed-time graph mean?

The speed is constant, so the acceleration is zero. The object is still moving, and the area under that section is still distance.

Q6How do you find the average speed for a whole journey?

Total distance divided by total time. It is the gradient of the line joining the first and last points of the graph.

Q7Why does a conversion graph pass through the origin?

Because zero of one unit is zero of the other, so the two quantities are in direct proportion.

Q8A conversion graph passes through

(5, 8)

. What is the conversion factor?

1.6, because

8 \div 5 = 1.6

. Multiply to go from the horizontal unit to the vertical one, and divide to come back.

Q9What does the intercept of a fare-distance graph represent?

The fixed charge: the amount paid when the distance is zero.

Q10How do you find the distance travelled from a speed-time graph made of straight sections?

Work out the area under the graph, splitting it into triangles, rectangles and trapezia, and add the parts.

0 of 10 marked as known

Exam-style questions

7 questions, 17 marks. Write your answer, then reveal the mark scheme and tick each point you actually made. Answering honestly here is what makes the rest of the site useful.

Write downAO12 marks

The graph of

y = 2 x - 3

is shown. Write down the gradient of the line and the value of

y

when

x = 0

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

B1gradient $= 2$ , because the line rises 2 for every 1 acrossnot checked yet
B1 $y = - 3$ when $x = 0$ not checked yet

Examiner tip: Write down means no working is needed. Read the gradient as the number of units the line rises for each 1 across, and the value at

x = 0

where the line crosses the

y

-axis.

Write downAO11 mark

A conversion graph for two units of length is a straight line passing through the origin and the point

(4, 10)

. Write down the gradient of the line.

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

B1 $10 \div 4 = 2.5$ not checked yet

Examiner tip: A conversion graph passes through the origin, so the gradient is simply the vertical coordinate divided by the horizontal one: 10 divided by 4.

Work outAO22 marks

A conversion graph converts miles to kilometres. The line passes through the origin and the point

(5, 8)

, so 5 miles is 8 kilometres. Work out the number of kilometres in 12 miles. Give your answer in kilometres.

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2. Include the unit (km).

Mark scheme — tick each point you actually made

M1the gradient is $8 \div 5 = 1.6$ kilometres per mile
A1 $12 \times 1.6 = 19.2$ kmnot checked yet

Examiner tip: Give the unit: the question asks for kilometres, so 19.2 on its own is an incomplete answer. The conversion factor is the gradient of the line, and 1.6 is worth remembering as the number of kilometres in a mile.

Work outAO23 marks

A cyclist's distance-time graph is a straight line from

(0, 0)

(30, 12)

, with the time in minutes and the distance in kilometres. Work out the cyclist's speed during those 30 minutes, in kilometres per hour.

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2. Include the unit (km/h).

Mark scheme — tick each point you actually made

M1the gradient is $12 \div 30 = 0.4$ kilometres per minute
M1multiplies by 60 to convert minutes into hours: $0.4 \times 60$
A124 km/hnot checked yet

Examiner tip: The gradient is in kilometres per minute, so it must be multiplied by 60 before it is a speed in km/h. An answer of 0.4 scores the first method mark and loses the rest.

Work outAO22 marks

A train's speed-time graph rises in a straight line from

(0, 0)

(20, 15)

, with the speed in metres per second and the time in seconds. Work out the acceleration during those 20 seconds. Give your answer in m/s

^{2}

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2. Include the unit (m/s^2).

Mark scheme — tick each point you actually made

M1acceleration is the gradient: $15 \div 20$
A1 $0.75$ m/s $^{2}$ not checked yet

Examiner tip: Acceleration is the gradient of a speed-time graph, not the area. Divide the speed by the time and not the time by the speed: 15 m/s gained in 20 s is 0.75 m/s

^{2}

Work outAO23 marks

A van's speed-time graph has three straight sections. The speed rises from 0 to 20 m/s in 30 seconds, stays at 20 m/s for 90 seconds, then falls to 0 in 20 seconds. Work out the total distance travelled. Give your answer in metres.

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2. Include the unit (m).

Mark scheme — tick each point you actually made

M1distance is the area under the graph, split into two triangles and a rectangle
M1 $\frac{1}{2} \times 30 \times 20 = 300$ , $20 \times 90 = 1800$ and $\frac{1}{2} \times 20 \times 20 = 200$
A1 $300 + 1800 + 200 = 2300$ mnot checked yet

Examiner tip: Write the three areas separately before adding them. The areas are in metres because the axes are metres per second and seconds, and one arithmetic slip then costs a method mark instead of the whole question.

Work outAO24 marks

A train leaves a station at 09:00 and travels at a constant 60 km/h for 40 minutes. It then waits at a second station for 10 minutes before returning to the first station at a constant 80 km/h. Work out the time at which it arrives back at the first station.

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

M1distance to the second station $= 60 \times \frac{40}{60} = 40$ km
M1return time $= 40 \div 80 = 0.5$ hours, which is 30 minutes
M1total time from 09:00 is $40 + 10 + 30 = 80$ minutes
A109:00 plus 80 minutes is 10:20not checked yet

Examiner tip: Sketch the journey as a distance-time graph: a line up for 40 minutes, a horizontal wait, then a steeper line back. Each leg gives one mark, and the final answer has to be a time rather than a number of minutes.

Exam technique for this topic

Write the units of the gradient before you calculate it. On a distance-time graph with kilometres and hours the gradient is in km/h, so if you worked in minutes multiply by 60 before writing the answer down.
Decide what a speed-time graph question wants before you start: the gradient gives acceleration and the area gives distance. Underline the word acceleration, distance or area in the question, because that one word chooses the whole method.
For an area under a speed-time graph, draw the vertical lines that split the region into triangles, rectangles and trapezia, then write each area on the diagram. The method mark is for the split and the separate areas, so a single wrong area costs one mark rather than all of them.
When a journey has more than one stage, answer in stages and keep the times as they appear on the axis. Convert everything into hours at the start if you are working in km/h, and write the conversion next to the graph so you do not mix minutes with hours in the final line.

Why students lose marks here

A student wroteThe gradient of a distance-time graph is the distance travelled.

Why it loses marksThe gradient is the rate at which the distance changes, which is the speed. The distance is read from the vertical axis at a particular time; it is not the gradient.

The answer that scoresGradient = change in distance divided by change in time = speed, in km/h or m/s. On a graph through

(0, 0)

and

(30, 12)

the speed is

12 \div 30 = 0.4

km per minute, which is 24 km/h.

A student wroteThe area under a speed-time graph is the acceleration.

Why it loses marksArea and gradient are two different measurements of the same graph. On a speed-time graph the gradient is the acceleration and the area is the distance travelled.

The answer that scoresSplit the area under the graph into triangles and rectangles and add them. For a graph that rises from 0 to 12 m/s in 8 s, stays at 12 m/s for 20 s and falls to 0 in 6 s, the distance is

48 + 240 + 36 = 324

A student wroteA horizontal section of a speed-time graph means the object is stationary.

Why it loses marksA horizontal line means the speed is constant, so the acceleration is zero. The object is still moving, and the area under that section is still distance travelled.

The answer that scoresHorizontal on a speed-time graph means a steady speed. Horizontal on a distance-time graph means stationary, because the distance is not changing.

A student wroteThe gradient is the change in

x

divided by the change in

y

Why it loses marksThe gradient is the change in the vertical quantity divided by the change in the horizontal one. Inverting it gives the reciprocal, which is a different rate: minutes per kilometre instead of kilometres per minute.

The answer that scoresGradient = change in

y

divided by change in

x

. A graph of distance against time therefore gives kilometres per hour, not hours per kilometre.

Where this fits in IGCSE Maths

This page is one topic of Cambridge IGCSE Maths 0580. IGCSE Maths is the course home: every area and every topic in Cambridge's own order, the papers you sit and what each one tests, and the exam-practice tools. The full topic list, with each topic linked to its revision page, is on the IGCSE Maths syllabus page.